Theorem al0ssb 5308

Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
al0ssb	⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Distinct variable group:   𝑦,𝑋

Proof of Theorem al0ssb

Step	Hyp	Ref	Expression
1		0ex 5307	. . 3 ⊢ ∅ ∈ V
2		sseq2 4010	. . . 4 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ ∅))
3		ss0b 4401	. . . 4 ⊢ (𝑋 ⊆ ∅ ↔ 𝑋 = ∅)
4	2, 3	bitrdi 287	. . 3 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅))
5	1, 4	spcv 3605	. 2 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 → 𝑋 = ∅)
6		0ss 4400	. . . 4 ⊢ ∅ ⊆ 𝑦
7	6	ax-gen 1795	. . 3 ⊢ ∀𝑦∅ ⊆ 𝑦
8		sseq1 4009	. . . 4 ⊢ (𝑋 = ∅ → (𝑋 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦))
9	8	albidv 1920	. . 3 ⊢ (𝑋 = ∅ → (∀𝑦 𝑋 ⊆ 𝑦 ↔ ∀𝑦∅ ⊆ 𝑦))
10	7, 9	mpbiri 258	. 2 ⊢ (𝑋 = ∅ → ∀𝑦 𝑋 ⊆ 𝑦)
11	5, 10	impbii 209	1 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)

Syntax hints:   ↔ wb 206  ∀wal 1538   = wceq 1540   ⊆ wss 3951  ∅c0 4333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334

This theorem is referenced by:  iota0def  47050  aiota0def  47108